Simplify and expand the following expression: $ \dfrac{p + 1}{5p - 3}+\dfrac{p}{5p - 6} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5p - 3)(5p - 6)$ Multiply the first term by $\dfrac{5p - 6}{5p - 6}$ $ \begin{align*} \dfrac{p + 1}{5p - 3} \times \dfrac{5p - 6}{5p - 6} & = \dfrac{(p + 1)(5p - 6)}{(5p - 3)(5p - 6)} \\ & = \dfrac{5p^2 - p - 6}{(5p - 3)(5p - 6)}\end{align*} $ Multiply the second term by $\dfrac{5p - 3}{5p - 3}$ $ \begin{align*} \dfrac{p}{5p - 6} \times \dfrac{5p - 3}{5p - 3} & = \dfrac{(p)(5p - 3)}{(5p - 6)(5p - 3)} \\ & = \dfrac{5p^2 - 3p}{(5p - 6)(5p - 3)}\end{align*} $ Now we have: $ = \dfrac{5p^2 - p - 6}{(5p - 3)(5p - 6)} + \dfrac{5p^2 - 3p}{(5p - 6)(5p - 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5p^2 - p - 6 + 5p^2 - 3p}{(5p - 3)(5p - 6)} $ $ = \dfrac{10p^2 - 4p - 6}{(5p - 3)(5p - 6)}$ Expand the denominator: $ = \dfrac{10p^2 - 4p - 6}{25p^2 - 45p + 18}$